By a result of Kaneko-Zagier (see [1]}, Proposition 1, part b), we know that the graded ring \(\widetilde{M}(\Gamma _1(3))\)  of quasi-modular forms on
\(\Gamma _1(3)\)  is given by
\(\widetilde{M}(\Gamma _1(3))=M(\Gamma _1(3))\otimes \mathbb {C}[e_2]\simeq \mathbb {C}[\vartheta ,\psi ^3,e_2]\, ,\) 
